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Statistics

Data Science Dojo
Syed Hanzala Ali
| October 16

“Statistics is the grammar of science”, Karl Pearson

In the world of data science, there is a secret language that benefits those who understand it. Do you want to know what makes a data expert efficient? It’s having a profound understanding of the data. Unfortunately, you can’t have a friendly conversation with the data, but don’t worry, we have the next best solution.

Here are the top ten statistical concepts that you must have in your arsenal.  Whether you’re a budding data scientist, a seasoned professional, or merely intrigued by the inner workings of data-driven decision-making, prepare for an enthralling exploration of the statistical principles that underpin the world of data science. 

 

 10 statistical concepts you should know

top statistical concepts
Top statistical concepts – Data Science Dojo

 

1. Descriptive statistics: 

Starting with the most fundamental and essential statistical concept, descriptive statistics. Descriptive statistics are the specific methods and measures that describe the data. It’s like the foundation of your building. It is a sturdy groundwork upon which further analysis can be constructed. Descriptive statistics can be broken down into measures of central tendency and measures of variability. 

  • Measure of Central Tendency: 

Central Tendency is defined as “the number used to represent the center or middle of a set of data values”. It is a single value that is typically representative of the whole data. They help us understand where the “average” or “central” point lies amidst a collection of data points.

There are a few techniques to find the central tendency of the data, namely “Mean” (average), “Median” (middle value when data is sorted), and “Mode” (most frequently occurring values).  

  • Measures of variability: 

Measures of variability describe the spread, dispersion, and deviation of the data. In essence, they tell us how much each value point deviates from the central tendency. A few measures of variability are “Range”, “Variance”, “Standard Deviation”, and “Quartile Range”. These provide valuable insights into the degree of variability or uniformity in the data.   

 

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 2. Inferential statistics: 

Inferential statistics enable us to draw conclusions about the population from a sample of the population. Imagine having to decide whether a medicinal drug is good or bad for the general public. It is practically impossible to test it on every single member of the population.

This is where inferential statistics comes in handy. Inferential statistics employ techniques such as hypothesis testing and regression analysis (also discussed later) to determine the likelihood of observed patterns occurring by chance and to estimate population parameters.

This invaluable tool empowers data scientists and researchers to go beyond descriptive analysis and uncover deeper insights, allowing them to make data-driven decisions and formulate hypotheses about the broader context from which the data was sampled. 

 

3. Probability distributions: 

Probability distributions serve as foundational concepts in statistics and mathematics, providing a structured framework for characterizing the probabilities of various outcomes in random events. These distributions, including well-known ones like the normal, binomial, and

Poisson distributions offer structured representations for understanding how data is distributed across different values or occurrences.

Much like navigational charts guiding explorers through uncharted territory, probability distributions function as reliable guides through the landscape of uncertainty, enabling us to quantitatively assess the likelihood of specific events.

They constitute essential tools for statistical analysis, hypothesis testing, and predictive modeling, furnishing a systematic approach to evaluate, analyze, and make informed decisions in scenarios involving randomness and unpredictability. Comprehension of probability distributions is imperative for effectively modeling and interpreting real-world data and facilitating accurate predictions. 

 

probability distributions 

Read More —-> 7 types of statistical distributions with practical examples 

 

4. Sampling methods: 

We now know inferential statistics help us make conclusions about the population from a sample of the population. How do we ensure that the sample is representative of the population? This is where sampling methods come to aid us.

Sampling methods are a set of methods that help us pick our sample set out of the population. Sampling methods are indispensable in surveys, experiments, and observational studies, ensuring that our conclusions are both efficient and statistically valid. There are many types of sampling methods. Some of the most common ones are defined below. 

  • Simple Random Sampling: A method where each member of the population has an equal chance of being selected for the sample, typically through random processes. 
  • Stratified Sampling: The population is divided into subgroups (strata), and a random sample is taken from each stratum in proportion to its size. 
  • Systematic Sampling: Selecting every “kth” element from a population list, using a systematic approach to create the sample. 
  • Cluster Sampling: The population is divided into clusters, and a random sample of clusters is selected, with all members in selected clusters included. 
  • Convenience Sampling: Selection of individuals/items based on convenience or availability, often leading to non-representative samples. 
  • Purposive (Judgmental) Sampling: Researchers deliberately select specific individuals/items based on their expertise or judgment, potentially introducing bias. 
  • Quota Sampling: The population is divided into subgroups, and individuals are purposively selected from each subgroup to meet predetermined quotas. 
  • Snowball Sampling: Used in hard-to-reach populations, where participants refer researchers to others, leading to an expanding sample. 

 

5. Regression analysis: 

Regression analysis is a statistical method that helps us quantify the relationship between a dependent variable and one or more independent variables. It’s like drawing a line through data points to understand and predict how changes in one variable relate to changes in another.

Regression models, such as linear regression or logistic regression, are used to uncover patterns and causal relationships in diverse fields like economics, healthcare, and social sciences. This technique empowers researchers to make predictions, analyze cause-and-effect connections, and gain insights into complex phenomena. 

 

Learn practical data science today!

 

6. Hypothesis testing: 

Hypothesis testing is a key statistical method used to assess claims or hypotheses about a population using sample data. It’s like a process of weighing evidence to determine if there’s enough proof to support a hypothesis.

Researchers formulate a null hypothesis and an alternative hypothesis, then use statistical tests to evaluate whether the data supports rejecting the null hypothesis in favor of the alternative.

This method is crucial for making informed decisions, drawing meaningful conclusions, and assessing the significance of observed effects in various fields of research and decision-making. 

 

7. Data visualizations: 

Data visualization is the art and science of representing complex data in a visual and comprehensible form. It’s like translating the language of numbers and statistics into a graphical story that anyone can understand at a glance.

Effective data visualization not only makes data more accessible but also allows us to spot trends, patterns, and outliers, making it an essential tool for data analysis and decision-making. Whether through charts, graphs, maps, or interactive dashboards, data visualization empowers us to convey insights, share information, and gain a deeper understanding of complex datasets. 

 

data science plots
9 Data Science Plots

 

Check out some of the most important plots for Data Science here. 

 

8. ANOVA (Analysis of variance): 

Analysis of Variance (ANOVA) is a statistical technique used to compare the means of two or more groups to determine if there are significant differences among them. It’s like the referee in a sports tournament, checking if there’s enough evidence to conclude that the teams’ performances are different.

ANOVA calculates a test statistic and a p-value, which indicates whether the observed differences in means are statistically significant or likely occurred by chance.

This method is widely used in research and experimental studies, allowing researchers to assess the impact of different factors or treatments on a dependent variable and draw meaningful conclusions about group differences. ANOVA is a powerful tool for hypothesis testing and plays a vital role in various fields, from medicine and psychology to economics and engineering. 

 

9. Time Series analysis: 

Time series analysis is a specialized field of statistics and data science that focuses on studying data points collected, recorded, or measured over time. It’s like examining the historical trajectory of a variable to understand its patterns and trends.

Time series analysis involves techniques for data visualization, smoothing, forecasting, and modeling to uncover insights and make predictions about future values.

This discipline finds applications in various domains, from finance and economics to climate science and stock market predictions, helping analysts and researchers understand and harness the temporal patterns within their data. 

 

10. Bayesian statistics: 

Bayesian statistics is a branch of statistics that takes a unique approach to probability and inference. Unlike classical statistics, which use fixed parameters, Bayesian statistics treat probability as a measure of uncertainty, updating beliefs based on prior information and new evidence.

It’s like continually refining your knowledge as you gather more data. Bayesian methods are particularly useful when dealing with complex, uncertain, or small-sample data, and they have applications in fields like machine learning, Bayesian networks, and decision analysis 

 

Ali Haider - Author
Ali Haider Shalwani
| October 8

In the realm of data science, understanding probability distributions is crucial. They provide a mathematical framework for modeling and analyzing data.  

 

Understand the applications of probability in data science with this blog.  

9 probability distributions in data science
9 probability distributions in data science – Data Science Dojo


Explore probability distributions in data science with practical applications

This blog explores nine important data science distributions and their practical applications. 

 

1. Normal distribution

The normal distribution, characterized by its bell-shaped curve, is prevalent in various natural phenomena. For instance, IQ scores in a population tend to follow a normal distribution. This allows psychologists and educators to understand the distribution of intelligence levels and make informed decisions regarding education programs and interventions.  

Heights of adult males in a given population often exhibit a normal distribution. In such a scenario, most men tend to cluster around the average height, with fewer individuals being exceptionally tall or short. This means that the majority fall within one standard deviation of the mean, while a smaller percentage deviates further from the average. 

 

2. Bernoulli distribution

The Bernoulli distribution models a random variable with two possible outcomes: success or failure. Consider a scenario where a coin is tossed. Here, the outcome can be either a head (success) or a tail (failure). This distribution finds application in various fields, including quality control, where it’s used to assess whether a product meets a specific quality standard. 

When flipping a fair coin, the outcome of each flip can be modeled using a Bernoulli distribution. This distribution is aptly suited as it accounts for only two possible results – heads or tails. The probability of success (getting a head) is 0.5, making it a fundamental model for simple binary events. 

 

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3. Binomial distribution

The binomial distribution describes the number of successes in a fixed number of Bernoulli trials. Imagine conducting 10 coin flips and counting the number of heads. This scenario follows a binomial distribution. In practice, this distribution is used in fields like manufacturing, where it helps in estimating the probability of defects in a batch of products. 

Imagine a basketball player with a 70% free throw success rate. If this player attempts 10 free throws, the number of successful shots follows a binomial distribution. This distribution allows us to calculate the probability of making a specific number of successful shots out of the total attempts. 

 

4. Poisson distribution

The Poisson distribution models the number of events occurring in a fixed interval of time or space, assuming a constant rate. For example, in a call center, the number of calls received in an hour can often be modeled using a Poisson distribution. This information is crucial for optimizing staffing levels to meet customer demands efficiently. 

In the context of a call center, the number of incoming calls over a given period can often be modeled using a Poisson distribution. This distribution is applicable when events occur randomly and are relatively rare, like calls to a hotline or requests for customer service during specific hours. 

 

5. Exponential distribution

The exponential distribution represents the time until a continuous, random event occurs. In the context of reliability engineering, this distribution is employed to model the lifespan of a device or system before it fails. This information aids in maintenance planning and ensuring uninterrupted operation. 

The time intervals between successive earthquakes in a certain region can be accurately modeled by an exponential distribution. This is especially true when these events occur randomly over time, but the probability of them happening in a particular time frame is constant. 

 

6. Gamma distribution

The gamma distribution extends the concept of the exponential distribution to model the sum of k independent exponential random variables. This distribution is used in various domains, including queuing theory, where it helps in understanding waiting times in systems with multiple stages. 

Consider a scenario where customers arrive at a service point following a Poisson process, and the time it takes to serve them follows an exponential distribution. In this case, the total waiting time for a certain number of customers can be accurately described using a gamma distribution. This is particularly relevant for modeling queues and wait times in various service industries. 

 

7. Beta distribution

The beta distribution is a continuous probability distribution bound between 0 and 1. It’s widely used in Bayesian statistics to model probabilities and proportions. In marketing, for instance, it can be applied to optimize conversion rates on a website, allowing businesses to make data-driven decisions to enhance user experience. 

In the realm of A/B testing, the conversion rate of users interacting with two different versions of a webpage or product is often modeled using a beta distribution. This distribution allows analysts to estimate the uncertainty associated with conversion rates and make informed decisions regarding which version to implement. 

 

8. Uniform distribution

In a uniform distribution, all outcomes have an equal probability of occurring. A classic example is rolling a fair six-sided die. In simulations and games, the uniform distribution is used to model random events where each outcome is equally likely. 

When rolling a fair six-sided die, each outcome (1 through 6) has an equal probability of occurring. This characteristic makes it a prime example of a discrete uniform distribution, where each possible outcome has the same likelihood of happening. 

 

9. Log normal distribution

The log normal distribution describes a random variable whose logarithm is normally distributed. In finance, this distribution is applied to model the prices of financial assets, such as stocks. Understanding the log normal distribution is crucial for making informed investment decisions. 

The distribution of wealth among individuals in an economy often follows a log-normal distribution. This means that when the logarithm of wealth is considered, the resulting values tend to cluster around a central point, reflecting the skewed nature of wealth distribution in many societies. 

 

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Learn probability distributions today! 

Understanding these distributions and their applications empowers data scientists to make informed decisions and build accurate models. Remember, the choice of distribution greatly impacts the interpretation of results, so it’s a critical aspect of data analysis. 

Delve deeper into probability with this short tutorial 

 

 

 

Author image - Ayesha
Ayesha Saleem
| September 19

The world we live in is defined by numbers and equations. From the simplest calculations to the most complex scientific theories, equations are the threads that weave the fabric of our understanding.

In this blog, we will step on a journey through the corridors of mathematical and scientific history, where we encounter the most influential equations that have shaped the course of human knowledge and innovation.

These equations are not mere symbols on a page; they are the keys that unlocked the mysteries of the universe, allowed us to build bridges that span great distances, enabled us to explore the cosmos, and even predicted the behavior of financial markets.

Get into the worlds of geometry, physics, mathematics, and more, to uncover the stories behind these 17 equations. From Pythagoras’s Theorem to the Black-Scholes Equation, each has its own unique tale, its own moment of revelation, and its own profound impact on our lives.

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Geometry and trigonometry:


1. Pythagoras’s theorem

Formula: a^2 + b^2 = c^2

Pythagoras’s Theorem is a mathematical formula that relates the lengths of the three sides of a right triangle. It states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Example:

Suppose you have a right triangle with two sides that measure 3 cm and 4 cm. To find the length of the hypotenuse, you would use the Pythagorean Theorem:

a^2 + b^2 = c^2

3^2 + 4^2 = c^2

9 + 16 = c^2

25 = c^2

c = 5

Therefore, the hypotenuse of the triangle is 5 cm.

Pythagoras’s Theorem is used in many different areas of work, including construction, surveying, and engineering. It is also used in everyday life, such as when measuring the distance between two points or calculating the height of a building.

Mathematics:

2. Logarithms

Formula: log(a, b) = c

Logarithms are a mathematical operation that is used to solve exponential equations. They are also used to scale numbers and compress data.

Example: Suppose you want to find the value of x in the following equation:2^x = 1024You can use logarithms to solve this equation by taking the logarithm of both sides:log(2^x, 2) = log(1024, 2)x * log(2, 2) = 10 * log(2, 2)x = 10Therefore, the value of x is 10.Logarithms are used in many different areas of work, including finance, engineering, and science.

They are also used in everyday life, such as when calculating interest rates or converting units.

3. Calculus

Calculus is a branch of mathematics that deals with rates of change. It is used to solve problems in many different areas of work, including physics, engineering, and economics.

One of the most important concepts in calculus is the derivative. The derivative of a function measures the rate of change of the function at a given point.

Another important concept in calculus is the integral. The integral of a function is the sum of the infinitely small areas under the curve of the function.

Example:

Suppose you have a function that represents the distance you have traveled over time. The derivative of this function would represent your speed. The integral of this function would represent your total distance traveled.

Calculus is a powerful tool that can be used to solve many different types of problems. It is used in many different areas of work, including science, engineering, and economics.

4. Chaos theory

Chaos theory is a branch of mathematics that studies the behavior of dynamic systems. It is used to model many different types of systems, such as the weather, the stock market, and the human heart.

One of the most important concepts in chaos theory is the butterfly effect. The butterfly effect states that small changes in the initial conditions of a system can lead to large changes in the long-term behavior of the system.

Example:

Suppose you have a butterfly flapping its wings in Brazil. This could cause a small change in the atmosphere, which could eventually lead to a hurricane in Florida.

Chaos theory is used in many different areas of physics, engineering, and economics. It is also used in everyday life, such as when predicting the weather and managing financial risks.

Learn about the Top 7 Statistical Techniques

Physics:

5. Law of gravity

Formula: F = G * (m1 * m2) / r^2

The law of gravity is a physical law that describes the gravitational force between two objects. It states that the force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them.

Example:

Suppose you have two objects, each with a mass of 1 kg. The gravitational force between the two objects would be 6.67 x 10^-11 N.

If you double the distance between the two objects, the gravitational force between them would be halved.

The law of gravity is used in many different areas of work, including astronomy, space exploration, and engineering. It is also used in everyday life, such as when calculating the weight of an object or the trajectory of a projectile.

Complex Numbers

6.The square root of minus one

Formula: i = sqrt(-1)

The square root of minus one is a complex number that is denoted by the letter i. It is defined as the number that, when multiplied by itself, equals -1.

Example:

i * i = -1

The square root of minus one is used in many different areas of mathematics, physics, and engineering. It is also used in everyday life, such as when calculating the voltage and current in an electrical circuit.

Read the Top 10 Statistics Books for Data Science

Geometry and Topology

7. Euler’s formula for Polyhedra

Formula: V – E + F = 2

Euler’s formula for polyhedra is a mathematical formula that relates the number of vertices, edges, and faces of a polyhedron. It states that the number of vertices minus the number of edges plus the number of faces is always equal to 2.

Example:

Suppose you have a cube. A cube has 8 vertices, 12 edges, and 6 faces. If you plug these values into Euler’s formula, you get:

V – E + F = 2

8 – 12 + 6 = 2

Therefore, Euler’s formula is satisfied.

Statistics and Probability:

8. Normal distribution

Formula: f(x) = exp(-(x-mu)^2/(2sigma^2)) / sqrt(2pi*sigma^2)
The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetrical and bell-shaped. It is used to model many different natural phenomena, such as human height, IQ scores, and measurement errors.
Example: Suppose you have a class of 30 students, and you want to know the average height of the students. You measure the height of each student and create a histogram of the results. You will likely find that the histogram is bell-shaped, with most of the students clustered around the average height and fewer students at the extremes.
This is because the height of humans is normally distributed. The normal distribution is used in many different areas of work, including statistics, finance, and engineering. It is also used in everyday life, such as when predicting the likelihood of a certain event happening.

9. Information theory

Formula: H(X) = -∑p(x) log2(p(x))

Information theory is a branch of mathematics that studies the transmission and processing of information. It was developed by Claude Shannon in the mid-20th century.

One of the most important concepts in information theory is entropy. Entropy is a measure of the uncertainty in a message. The higher the entropy of a message, the more uncertain it is.

Example:

Suppose you have a coin. The entropy of the coin is 1 bit, because there are two equally likely outcomes: heads or tails.

If you flip the coin and it lands on heads, the entropy of the coin is 0, because there is only one possible outcome: heads.

Information theory is used in many different areas of communication, computer science, and statistics. It is also used in everyday life, such as when designing data compression algorithms and communication protocols.

Physics and Engineering:

10. Wave equation

Formula: ∂^2u/∂t^2 = c^2 * ∂^2u/∂x^2

The wave equation is a differential equation that describes the propagation of waves. It is used to model many different types of waves, such as sound waves, light waves, and water waves.

Example:

Suppose you throw a rock into a pond. The rock will create a disturbance in the water that will propagate outwards in the form of a wave. The wave equation can be used to model the propagation of this wave.

The wave equation is used in many different areas of physics, engineering, and computer science. It is also used in everyday life, such as when designing sound systems and optical devices.

Learn about Top Machine Learning Algorithms for Data Science

11. Fourier transform

Formula: F(u) = ∫ f(x) * exp(-2pii*ux) dx

The Fourier transform is a mathematical operation that transforms a function from the time domain to the frequency domain. It is used to analyze signals and images.

Example:

Suppose you have a sound recording. The Fourier transform of the sound recording can be used to identify the different frequencies that are present in the recording. This information can then be used to compress the recording or to remove noise from the recording.

The Fourier transform is used in many different areas of science and engineering. It is also used in everyday life, such as in digital signal processing and image processing.

12. Navier-Stokes equation

Formula: ρ * (∂u/∂t + (u ⋅ ∇)u) = -∇p + μ∇^2u + F

The Navier-Stokes equations are a system of differential equations that describe the motion of fluids. They are used to model many different types of fluid flow, such as the flow of air around an airplane wing and the flow of blood through the body.

Example:

Suppose you are designing an airplane wing. You can use the Navier-Stokes equations to simulate the flow of air around the wing and to determine the lift and drag forces that the wing will experience.

The Navier-Stokes equations are used in many different areas of engineering, such as aerospace engineering, mechanical engineering, and civil engineering. They are also used in physics and meteorology.

13. Maxwell’s equations

Formula: ∇⋅E = ρ/ε0 | ∇×E = -∂B/∂t | ∇⋅B = 0 | ∇×B = μ0J + μ0ε0∂E/∂t

Maxwell’s equations are a set of four equations that describe the behavior of electric and magnetic fields. They are used to model many different phenomena, such as the propagation of light waves and the operation of electrical devices.

Example:

Suppose you are designing a generator. You can use Maxwell’s equations to simulate the flow of electric and magnetic fields in the generator and to determine the amount of electricity that the generator will produce.

Maxwell’s equations are used in many different areas of physics and engineering. They are also used in everyday life, such as in the design of electrical devices and communication systems.

14. Second Law of thermodynamics

Formula: dS ≥ 0

The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. Entropy is a measure of the disorder of a system.

Example:

Suppose you have a cup of hot coffee. The coffee is initially ordered, with the hot molecules at the top of the cup and the cold molecules at the bottom of the cup. Over time, the coffee will cool down and the molecules will become more disordered. This is because the second law of thermodynamics requires the total entropy of the system to increase over time.

The second law of thermodynamics is used in many different areas of physics, engineering, and economics. It is also used in everyday life, such as when designing power plants and refrigerators.

Physics and Cosmology:

15. Relativity

Formula: E = mc^2

Relativity is a branch of physics that studies the relationship between space and time. It was developed by Albert Einstein in the early 20th century. One of the most famous equations in relativity is E = mc^2, which states that energy and mass are equivalent. This means that energy can be converted into mass and vice versa.

Example: Suppose you have a nuclear reactor. The nuclear reactor converts nuclear energy into mass. This is because the nuclear reactor converts the energy of the nuclear binding force into mass. Relativity is used in many different areas of physics, astronomy, and engineering. It is also used in everyday life, such as in the design of GPS systems and particle accelerators.

16. Schrödinger’s equation

Formula: iℏ∂ψ/∂t = Hψ

Schrödinger’s equation is a differential equation that describes the behavior of quantum mechanical systems. It is used to model many different types of quantum systems, such as atoms, molecules, and electrons.

Example:

Suppose you have a hydrogen atom. The Schrödinger equation can be used to calculate the energy levels of the hydrogen atom and the probability of finding the electron in a particular region of space.

Schrödinger’s equation is used in many different areas of physics, chemistry, and materials science. It is also used in the development of new technologies, such as quantum computers and quantum lasers.

Finance and Economics:

17. Black-Scholes equation

Formula: ∂C/∂t + ½σ^2S^2∂^2C/∂S^2 – rC = 0

The Black-Scholes equation is a differential equation that describes the price of a European option. A European option is a financial contract that gives the holder the right, but not the obligation, to buy or sell an asset at a certain price on a certain date.

The Black-Scholes equation is used to price options and to develop hedging strategies. It is one of the most important equations in finance.

Example:

Suppose you are buying a call option on a stock. The Black-Scholes equation can be used to calculate the price of the call option. This information can then be used to decide whether or not to buy the call option and to determine how much to pay for it.

The Black-Scholes equation is used by many different financial institutions, such as investment banks and hedge funds. It is also used by individual investors to make investment decisions.

Share your favorite equation with us!

Mathematics and science are not just abstract concepts but the very foundations upon which our modern world stands. These 17 equations have not only changed the way we see the world but have also paved the way for countless innovations and advancements.

From the elegance of Euler’s Formula for Polyhedra to the complexity of Maxwell’s Equations, from the order of Normal Distribution to the chaos of Chaos Theory, each equation has left an indelible mark on the human story.

They have transcended their origins and become tools that shape our daily lives, drive technological progress, and illuminate the mysteries of the cosmos.

As we continue to explore, learn, and discover, let us always remember the profound impact of these equations and the brilliant minds behind them. They remind us that the pursuit of knowledge knows no bounds and that the world of equations is a realm of infinite wonder and possibility.

Let us know in the comments in case we missed any!

Ruhma Khawaja author
Ruhma Khawaja
| May 12

Probability is a fundamental concept in data science. It provides a framework for understanding and analyzing uncertainty, which is an essential aspect of many real-world problems. In this blog, we will discuss the importance of probability in data science, its applications, and how it can be used to make data-driven decisions. 

What is probability? 

It is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain. For example, the probability of rolling a six on a fair die is 1/6 or approximately 0.17. 

In data science, it is used to quantify the uncertainty associated with data. It helps data scientists to make informed decisions by providing a way to model and analyze the variability of data. It is also used to build models that can predict future events or outcomes based on past data. 

Applications of probability in data science 

There are many applications of probability in data science, some of which are discussed below: 

1. Statistical inference:

Statistical inference is the process of drawing conclusions about a population based on a sample of data. It plays a central role in statistical inference by providing a way to quantify the uncertainty associated with estimates and hypotheses. 

2. Machine learning:

Machine learning algorithms make predictions about future events or outcomes based on past data. For example, a classification algorithm might use probability to determine the likelihood that a new observation belongs to a particular class. 

3. Bayesian analysis:

Bayesian analysis is a statistical approach that uses probability to update beliefs about a hypothesis as new data becomes available. It is commonly used in fields such as finance, engineering, and medicine. 

4. Risk assessment:

It is used to assess risk in many industries, including finance, insurance, and healthcare. Risk assessment involves estimating the likelihood of a particular event occurring and the potential impact of that event. 

Applications of probability in data science 
Applications of probability in data science

5. Quality control:

It is used in quality control to determine whether a product or process meets certain specifications. For example, a manufacturer might use probability to determine whether a batch of products meets a certain level of quality.

6. Anomaly detection

Probability is used in anomaly detection to identify unusual or suspicious patterns in data. By modeling the normal behavior of a system or process using probability distributions, any deviations from the expected behavior can be detected as anomalies. This is valuable in various domains, including cybersecurity, fraud detection, and predictive maintenance.

How probability helps in making data-driven decisions 

It help data scientists to make data-driven decisions by providing a way to quantify the uncertainty associated with data. By using  to model and analyze data, data scientists can: 

  • Estimate the likelihood of future events or outcomes based on past data. 
  • Assess the risk associated with a particular decision or action. 
  • Identify patterns and relationships in data. 
  • Make predictions about future trends or behavior. 
  • Evaluate the effectiveness of different strategies or interventions. 

Bayes’ theorem and its relevance in data science 

Bayes’ theorem, also known as Bayes’ rule or Bayes’ law, is a fundamental concept in probability theory that has significant relevance in data science. It is named after Reverend Thomas Bayes, an 18th-century British statistician and theologian, who first formulated the theorem. 

At its core, Bayes’ theorem provides a way to calculate the probability of an event based on prior knowledge or information about related events. It is commonly used in statistical inference and decision-making, especially in cases where new data or evidence becomes available. 

The theorem is expressed mathematically as follows: 

P(A|B) = P(B|A) * P(A) / P(B) 

Where: 

  • P(A|B) is the probability of event A occurring given that event B has occurred. 
  • P(B|A) is the probability of event B occurring given that event A has occurred. 
  • P(A) is the prior probability of event A occurring. 
  • P(B) is the prior probability of event B occurring. 

In data science, Bayes’ theorem is used to update the probability of a hypothesis or belief in light of new evidence or data. This is done by multiplying the prior probability of the hypothesis by the likelihood of the new evidence given that hypothesis.

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For example, let’s say we have a medical test that can detect a certain disease, and we know that the test has a 95% accuracy rate (i.e., it correctly identifies 95% of people with the disease and 5% of people without it). We also know that the prevalence of the disease in the population is 1%. If we administer the test to a person and they test positive, we can use Bayes’ theorem to calculate the probability that they actually have the disease. 

In conclusion, Bayes’ theorem is a powerful tool for probabilistic inference and decision-making in data science. Incorporating prior knowledge and updating it with new evidence, it enables more accurate and informed predictions and decisions. 

Common mistakes to avoid in probability analysis 

Probability analysis is an essential aspect of data science, providing a framework for making informed predictions and decisions based on uncertain events. However, even the most experienced data scientists can make mistakes when applying probability analysis to real-world problems. In this article, we’ll explore some common mistakes to avoid: 

  • Assuming independence: One of the most common mistakes is assuming that events are independent when they are not. For example, in a medical study, we may assume that the likelihood of developing a certain condition is independent of age or gender, when in reality these factors may be highly correlated. Failing to account for such dependencies can lead to inaccurate results. 
  • Misinterpreting probability: Some people may think that a probability of 0.5 means that an event is certain to occur, when in fact it only means that the event has an equal chance of occurring or not occurring. Properly understanding and interpreting probability is essential for accurate analysis. 
  • Neglecting sample size: Sample size plays a critical role in probability analysis. Using a small sample size can lead to inaccurate results and incorrect conclusions. On the other hand, using an excessively large sample size can be wasteful and inefficient. Data scientists need to strike a balance and choose an appropriate sample size based on the problem at hand. 
  • Confusing correlation and causation: Another common mistake is confusing correlation with causation. Just because two events are correlated does not mean that one causes the other. Careful analysis is required to establish causality, which can be challenging in complex systems. 
  • Ignoring prior knowledge: Bayesian probability analysis relies heavily on prior knowledge and beliefs. Failing to consider prior knowledge or neglecting to update it based on new evidence can lead to inaccurate results. Properly incorporating prior knowledge is essential for effective Bayesian analysis. 
  • Overreliance on models: The models can be powerful tools for analysis, but they are not infallible. Data scientists need to exercise caution and be aware of the assumptions and limitations of the models they use. Blindly relying on models can lead to inaccurate or misleading results. 

Conclusion 

Probability is a powerful tool for data scientists. It provides a way to quantify uncertainty and make data-driven decisions. By understanding the basics of probability and its applications in data science, data scientists can build models and make predictions that are both accurate and reliable. As data becomes increasingly important in all aspects of our lives, the ability to use it effectively will become an essential skill for success in many fields. 

 

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Ayesha Saleem
| April 7

The Poisson process is a popular method of counting random events that occur at a certain rate. It is commonly used in situations where the timing of events appears to be random, but the rate of occurrence is known. For example, the frequency of earthquakes in a specific region or the number of car accidents at a location can be modeled using the Poisson process. 

It is a fundamental concept in probability theory that is widely used to model a range of phenomena where events occur randomly over time. Named after the French mathematician Siméon Denis Poisson, this stochastic process has applications in diverse fields such as physics, biology, engineering, and finance.

In this article, we will explore the mathematical definition of the Poisson process, its parameters and applications, as well as its limitations and extensions. We will also discuss the history and development of this concept and its significance in modern research.

Understanding the parameters of the poisson process

The Poisson process is defined by several key properties: 

  • Events happen at a steady rate over time 
  • The probability of an event happening in a short period of time is inversely proportional to the duration of the interval, and 
  • Events take place independently of one another.  


Additionally, the Poisson distribution governs the number of events that take place during a specific period, and the rate parameter (which determines the mean and variance) is the only parameter that can be used to describe it.
 

Defining poisson process
Defining poisson process

Mathematical definition of the poisson process

To calculate the probability of a given number of events occurring in a Poisson process, the Poisson distribution formula is used: P(x) = (lambda^x * e^(-lambda)) / x! where lambda is the rate parameter and x! is the factorial of x. 

The Poisson process can be applied to a wide range of real-world situations, such as the arrival of customers at a store, the number of defects in a manufacturing process, the number of calls received by a call center, the number of accidents at a particular intersection, and the number of emails received by a person in a given time period.  

It’s essential to keep in mind that the Poisson process is a stochastic process that counts the number of events that have occurred in a given interval of time, while the Poisson distribution is a discrete probability distribution that describes the likelihood of events with a Poisson process happening in a given time period 

Real scenarios to use the poisson process

The Poisson process is a popular counting method used in situations where events occur at a certain rate but are actually random and without a certain structure. It is frequently used to model the occurrence of events over time, such as the number of faults in a manufacturing process or the arrival of customers at a store. Some examples of real-life situations where the Poisson process can be applied include: 

  • The arrival of customers at a store or other business: The rate at which customers arrive at a store can be modeled using a Poisson process, with the rate parameter representing the average number of customers that arrive per unit of time. 
  • The number of defects in a manufacturing process: The rate at which defects occur in a manufacturing process can be modeled using a Poisson process, with the rate parameter representing the average number of defects per unit of time. 
  • The number of calls received by a call center: The rate at which calls are received by a call center can be modeled using a Poisson process, with the rate parameter representing the average number of calls per unit of time. 
  • The number of accidents at a particular intersection: The rate at which accidents occur at a particular intersection can be modeled using a Poisson process, with the rate parameter representing the average number of accidents per unit of time. 
  • The number of emails received by a person in a given time period: The rate at which emails are received by a person can be modeled using a Poisson process, with the rate parameter representing the average number of emails received per unit of time. 


It’s also used in other branches of probability and statistics, including the analysis of data from experiments involving a large number of trials and the study of queues.
 

 

Explore more about probability distributions and their applications in the Poisson process by checking out our related articles on probability theory and data analysis.

Putting it into perspective

In conclusion, the Poisson process is a popular counting method that is often used in situations where events occur at a certain rate but are actually completely random. It is defined by the recurrence of events throughout time and has several properties, including a steady rate of events over time, an inverse correlation between the probability of an event happening and the duration of the interval, and independence of events from one another.

The Poisson distribution is used to calculate the probability of a given number of events occurring in a given interval of time in a Poisson process. The Poisson process has many real-world applications, including modeling the arrival of customers at a store, the number of defects in a manufacturing process, the number of calls received by a call center, and the number of accidents at a particular intersection.

Overall, it is a useful tool in probability and statistics for analyzing data from experiments involving a large number of trials and studying queues.  

Data Science Dojo
Prasad D Wilagama
| March 17

In today’s digital age, with a plethora of tools available at our fingertips, researchers can now collect and analyze data with greater ease and efficiency. These research tools not only save time but also provide more accurate and reliable results. In this blog post, we will explore some of the essential research tools that every researcher should have in their toolkit.

From data collection to data analysis and presentation, this blog will cover it all. So, if you’re a researcher looking to streamline your work and improve your results, keep reading to discover the must-have tools for research success.

Revolutionize your research: The top 20 must-have research tools

Research requires various tools to collect, analyze and disseminate information effectively. Some essential research tools include search engines like Google Scholar, JSTOR, and PubMed, reference management software like Zotero, Mendeley, and EndNote, statistical analysis tools like SPSS, R, and Stata, writing tools like Microsoft Word and Grammarly, and data visualization tools like Tableau and Excel.  

Essential Research Tools for Researchers

1. Google Scholar – Google Scholar is a search engine for scholarly literature, including articles, theses, books, and conference papers.

2. JSTOR – JSTOR is a digital library of academic journals, books, and primary sources.

3.PubMedPubMed is a free search engine accessing primarily the MEDLINE database of references and abstracts on life sciences and biomedical topics. 

4. Web of Science: Web of Science is a citation index that allows you to search for articles, conference proceedings, and books across various scientific disciplines. 

5. Scopus – Scopus citation database that covers scientific, technical, medical, and social sciences literature. 

6. Zotero: Zotero is a free, open-source citation management tool that helps you organize your research sources, create bibliographies, and collaborate with others.

7. Mendeley – Mendeley is a reference management software that allows you to organize and share your research papers and collaborate with others.

8. EndNote – EndNoted is a software tool for managing bibliographies, citations, and references on the Windows and macOS operating systems. 

9. RefWorks – RefWorks is a web-based reference management tool that allows you to create and organize a personal database of references and generate citations and bibliographies.

10. Evernote – Evernote is a digital notebook that allows you to capture and organize your research notes, web clippings, and documents.

11. SPSS – SPSS is a statistical software package used for data analysis, data mining, and forecasting.

12. R – R is a free, open-source software environment for statistical computing and graphics.

13. Stata – Stata is a statistical software package that provides a suite of applications for data management and statistical analysis.

Other helpful tools for collaboration and organization include NVivo, Slack, Zoom, and Microsoft Teams. With these tools, researchers can effectively find relevant literature, manage references, analyze data, write research papers, create visual representations of data, and collaborate with peers. 

14. Excel – Excel is spreadsheet software used for organizing, analyzing, and presenting data.

15. Tableau – Tableau is a data visualization software that allows you to create interactive visualizations and dashboards.

16. NVivo – Nviva is a software tool for qualitative research and data analysis.

17. Slack – Slack is a messaging platform for team communication and collaboration.

18. Zoom – Zoom is a video conferencing software that allows you to conduct virtual meetings and webinars.

19. Microsoft Teams – Microsoft Teams is a collaboration platform that allows you to chat, share files, and collaborate with your team.

20. Qualtrics – Qualtrics is an online survey platform that allows researchers to design and distribute surveys, collect and analyze data, and generate reports.

Maximizing accuracy and efficiency with research tools

Research is a vital aspect of any academic discipline, and it is critical to have access to appropriate research tools to facilitate the research process. Researchers require access to various research tools and software to conduct research, analyze data, and report research findings. Some standard research tools researchers use include search engines, reference management software, statistical analysis tools, writing tools, and data visualization tools.

Specialized research tools are also available for researchers in specific fields, such as GIS software for geographers and geneticist gene sequence analysis tools. These tools help researchers organize data, collaborate with peers, and effectively present research findings.

It is crucial for researchers to choose the right tools for their research project, as these tools can significantly impact the accuracy and reliability of research findings.

Conclusion

Summing it up, researchers today have access to an array of essential research tools that can help simplify the research process. From data collection to analysis and presentation, these tools make research more accessible, efficient, and accurate. By leveraging these tools, researchers can improve their work and produce more high-quality research.

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Ayesha Saleem
| February 7

Get ahead in data analysis with our summary of the top 7 must-know statistical techniques. Master these tools for better insights and results.

While the field of statistical inference is fascinating, many people have a tough time grasping its subtleties. For example, some may not be aware that there are multiple types of inference and that each is applied in a different situation. Moreover, the applications to which inference can be applied are equally diverse.

For example, when it comes to assessing the credibility of a witness, we need to know how reliable the person is and how likely it is that the person is lying. Similarly, when it comes to making predictions about the future, it is important to factor in not just the accuracy of the forecast but also whether it is credible. 

 

Top statistical techniques
Top statistical techniques – Data Science Dojo

 

Counterfactual causal inference: 

Counterfactual causal inference is a statistical technique that is used to evaluate the causal significance of historical events. Exploring how historical events may have unfolded under small changes in circumstances allows us to assess the importance of factors that may have caused the event. This technique can be used in a wide range of fields such as economics, history, and social sciences. There are multiple ways of doing counterfactual inference, such as Bayesian Structural Modelling. 

  

Overparametrized models and regularization: 

Overparametrized models are models that have more parameters than the number of observations. These models are prone to overfitting and are not generalizable to new data. Regularization is a technique that is used to combat overfitting in overparametrized models. Regularization adds a penalty term to the loss function to discourage the model from fitting the noise in the data. Two common types of regularization are L1 and L2 regularization. 

  

Generic computation algorithms: 

Generic computation algorithms are a set of algorithms that can be applied to a wide range of problems. These algorithms are often used to solve optimization problems, such as gradient descent and conjugate gradient. They are also used in machine learning, such as support vector machines and k-means clustering. 

  

Robust inference: 

Robust inference is a technique that is used to make inferences that are not sensitive to outliers or extreme observations. This technique is often used in cases where the data is contaminated with errors or outliers. There are several robust statistical methods such as the median and the Huber M-estimator. 

 

Read about: Key statistical distributions with real life scenarios

 

Bootstrapping and simulation-based inference: 

Bootstrapping and simulation-based inference are techniques that are used to estimate the precision of sample statistics and to evaluate and compare models. Bootstrapping is a resampling technique that is used to estimate the sampling distribution of a statistic by resampling the data with replacement.

Simulation-based inference is a method that is used to estimate the sampling distribution of a statistic by generating many simulated samples from the model. 

  

Enroll in our Data Science Bootcamp to learn more about statistical techniques

 

Multilevel models: 

Multilevel models are a class of models that are used to account for the hierarchical structure of data. These models are often used in fields such as education, sociology, and epidemiology. They are also known as hierarchical linear models, mixed-effects models, or random coefficient models. 

  

Adaptive decision analysis: 

Adaptive Decision Analysis is a statistical technique that is used to make decisions under uncertainty. It involves modeling the decision problem, simulating the outcomes of the decision and updating the decision based on the new information. This method is often used in fields such as finance, engineering, and healthcare. 

 

Which statistical techniques are most used by you? 

This article discusses most of the statistical methods that are used in quantitative fields. These are often used to infer causal relationships between variables. 

The primary goal of any statistical way is to infer causality from observational data. It is usually difficult to achieve this goal for two reasons. First, observational data may be noisy and contaminated by errors. Second, variables are often correlated. To correctly infer causality, it is necessary to model these correlations and to account for any biases and confounding factors. 

As statistical techniques are often implemented using specific software packages, the implementations of each method often differ. This article first briefly describes the papers and software packages that are used in the following sections. It then describes the most common statistical techniques and the best practices that are associated with each technique. 

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Ayesha Saleem
| December 21

In this blog, we are going to learn the differences and similarities between linear regression and logistic regression. 

 

Regression is a statistical technique used in the fields of finance, investing, and other disciplines that aim to establish the nature and strength of the relationship between a single dependent variable (often represented by Y) and a number of independent variables (known as independent variables). 

 

linear regression vs logistic regression
Linear regression vs logistic regression – Data Science Dojo

 

Forecasting and prediction both require regression analysis. There is a lot of overlap between this and machine learning. This statistical approach is employed in a variety of industries, including 

Financial: Understanding stock price trends, making price predictions, and assessing insurance risk.

Marketing: Analyze the success of marketing initiatives and project product pricing and sales. 

Manufacturing: Assess the relationships between the variables that define a better engine and its performance. 

Medicine: To produce generic medications for ailments, forecast the various medication combinations. 

 

The most popular variation of this method is linear regression, which is also known as simple regression or ordinary least squares (OLS). Based on a line of best fit, linear regression determines the linear relationship between two variables.  

The slope of a straight line used to represent linear regression thus indicates how changing one variable effect changing another. In a linear regression connection, the value of one variable when the value of the other is zero is represented by the y-intercept. There are also non-linear regression models, although they are far more complicated. 

 

Terminologies used in regression analysis

Outliers 

The term “outlier” refers to an observation in a dataset that has an extremely high or very low value in comparison to the other observations, i.e., it does not belong to the population. 

  

Multicollinearity 

The independent variables are said to be multicollinear when there is a strong correlation between them. 

  

Heteroscedasticity 

Heteroscedasticity refers to the non-constant fluctuation between the target variable and the independent variable. 

  

Both under- and over-fit 

Overfitting may result from the usage of extraneous explanatory variables. Overfitting occurs when our algorithm performs admirably on the training set but falls short on the test sets. 

 

Linear regression 

In simple terms, linear regression is used to find a relationship between two variables: a Dependent variable (y) and an independent variable (X) with the help of a straight line. It also makes predictions for continuous or numeric variables such as sales, salary, age, and product price and shows us how the value of the dependent variable changes with the change in the value of an independent variable.   

Watch more videos on machine learning at Data Science Dojo 

 

Let’s say we have a dataset available consisting of house areas in square meters and their respective prices.    

As the change in area results in a change in price change of a house, we will put the area on the X-axis as an independent variable and the Price on the Y-axis as a dependent variable.   

On the chart, these data points would appear as a scatter plot, a set of points that may or may not appear to be organized along any line.   

Now using this data, we are required to predict the price of houses having the following areas:   

500, 2000, and 3500.   

After plotting these points, if a linear pattern is visible, sketch a straight line as the line of best fit.  

The best fit line we draw minimizes the distance between it and the observed data. Estimating this line is a key component of regression analysis that helps to infer the relationships between a dependent variable and an independent variable.   

  

Measures for linear regression   

  

To understand the amount of error that exists between different models in linear regression, we use metrics. Let’s discuss some of the evaluation measures for regression:   

  

  • Mean Absolute Error    

  

Mean absolute error measures the absolute difference between the predicted and actual values of the model. This metric is the average prediction error. Lower MAE values indicate a better fit.   

  

  • Root Mean Squared Error  

     

Root Mean Squared Error indicates how different the residuals are from zero. Residuals represent the difference between the observed and predicted value of the dependent variable.    

  

  • R-Squared Measure 

  

R squared Measure is the standard deviation of the residuals. The plot of the residuals shows the distance of the data points from the regression line. The root mean squared error squares the residuals, average the residuals, and takes the square root. RMSE measures the difference between the actual target from the predicted values.   

  

Lower RMSE values indicate shorter distances from the actual data point to the line and therefore a better fit. RMSE uses the same units as the dependent value. 

 

Logistic regression 

Additionally, logistic models can modify raw data streams to produce characteristics for various AI and machine learning methods. In reality, one of the often employed machine learning techniques for binary classification issues, or problems with two class values, includes logistic regression. These problems include predictions like “this or that,” “yes or no,” and “A or B.” 

 

Read about logistic regression in R in this blog

 

The probability of occurrences can also be estimated using logistic regression, which includes establishing a link between feature likelihood and outcome likelihood. In other words, it can be applied to categorization by building a model that links the number of hours of study to the likelihood that a student would pass or fail. 

 

Comparison of linear regression and logistic regression 

The primary distinction between logistic and linear regression is that the output of logistic regression is constant whereas the output of linear regression is continuousutilized.   

The outcome, or dependent variable, in logistic regression has just two possible values. However, the output of a linear regression is continuous, which means that there are an endless number of possible values for it. 

When the response variable is categorical, such as yes/no, true/false, and pass/fail, logistic regression is utilised. When the response variable is continuous, like hours, height, or weight, linear regression is utilised. 

Logistic regression and linear regression, for instance, can predict various outcomes depending on the information about the amount of time a student spent studying and the results of their exams. 

 

Curve, a visual representation of linear and logistic regression

Regression curves
Regression curves – Visual representation of linear regression and logistic regression

 

A straight line, often known as a regression line, is used to indicate linear regression. This line displays the expected score on “y” for each value of “x.” Additionally, the distance between the data points on the plot and the regression line reveals model flaws. 

In contrast, an S-shaped curve is revealed using logistic regression. Here, the orientation and steepness of the curve are affected by changes in the regression coefficients. So, it follows that a positive slope yields an S-shaped curve, but a negative slope yields a Z-shaped curve. 

 

Which one to use – Linear regression or logistic regression?

Regression analysis requires careful attention to the problem statement, which must be understood before proceeding. It seems sense to apply linear regression if the problem description mentions forecasts. If binary classification is included in the issue statement, logistic regression should be used. Similarly, we must assess each of our regression models in light of the problem statement. 

  

Enroll in Data Science Bootcamp to learn more about these ideas and advance your career today. 

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Ayesha Saleem
| December 8

Statistical distributions help us understand a problem better by assigning a range of possible values to the variables, making them very useful in data science and machine learning. Here are 6 types of distributions with intuitive examples that often occur in real-life data. 

In statistics, a distribution is simply a way to understand how a set of data points are spread over some given range of values.  

For example, distribution takes place when the merchant and the producer agree to sell the product during a specific time frame. This form of distribution is exhibited by the agreement reached between Apple and AT&T to distribute their products in the United States. 

 

types of probability distribution
Types of probability distribution – Data Science Dojo

 

Types of statistical distributions 

There are several statistical distributions, each representing different types of data and serving different purposes. Here we will cover several commonly used distributions. 

  1. Normal Distribution 
  2. t-Distribution 
  3. Binomial Distribution 
  4. Poisson Distribution 
  5. Uniform Distribution 

 

Pro-tip: Enroll in the data science bootcamp today and advance your learning 

 

1. Normal Distribution 

A normal distribution also known as “Gaussian Distribution” shows the probability density for a population of continuous data (for example height in cm for all NBA players). Also, it indicates the likelihood that any NBA player will have a particular height. Let’s say fewer players are much taller or shorter than usual; most are close to average height.  

The spread of the values in our population is measured using a metric called standard deviation. The Empirical Rule tells us that: 

  • 68.3% of the values will fall between1 standard deviation above and below the mean 
  • 95.5% of the values will fall between2 standard deviations above and below the mean 
  • 99.7% of the values will fall between3 standard deviations above and below the mean 

 

Let’s assume that we know that the mean height of all players in the NBA is 200cm and the standard deviation is 7cm. If Le Bron James is 206 cm tall, what proportion of NBA players is he taller than? We can figure this out! LeBron is 6cm taller than the mean (206cm – 200cm). Since the standard deviation is 7cm, he is 0.86 standard deviations (6cm / 7cm) above the mean. 

Our value of 0.86 standard deviations is called the z-score. This shows that James is taller than 80.5% of players in the NBA!  

This can be converted to a percentile using the probability density function (or a look-up table) giving us our answer. A probability density function (PDF) defines the random variable’s probability of coming within a distinct range of values. 

 

2. t-distribution 

A t-distribution is symmetrical around the mean, like a normal distribution, and its breadth is determined by the variance of the data. A t-distribution is made for circumstances where the sample size is limited, but a normal distribution works with a population. With a smaller sample size, the t-distribution takes on a broader range to account for the increased level of uncertainty. 

The number of degrees of freedom, which is determined by dividing the sample size by one, determines the curve of a t-distribution. The t-distribution tends to resemble a normal distribution as sample size and degrees of freedom increase because a bigger sample size increases our confidence in estimating the underlying population statistics. 

For example, suppose we deal with the total number of apples sold by a shopkeeper in a month. In that case, we will use the normal distribution. Whereas, if we are dealing with the total amount of apples sold in a day, i.e., a smaller sample, we can use the t distribution. 

 

3. Binomial distribution 

A Binomial Distribution can look a lot like a normal distribution’s shape. The main difference is that instead of plotting continuous data, it plots a distribution of two possible discrete outcomes, for example, the results from flipping a coin. Imagine flipping a coin 10 times, and from those 10 flips, noting down how many were “Heads”. It could be any number between 1 and 10. Now imagine repeating that task 1,000 times. 

If the coin, we are using is indeed fair (not biased to heads or tails) then the distribution of outcomes should start to look at the plot above. In the vast majority of cases, we get 4, 5, or 6 “heads” from each set of 10 flips, and the likelihood of getting more extreme results is much rarer! 

 

4. Bernoulli distribution 

The Bernoulli Distribution is a special case of Binomial Distribution. It considers only two possible outcomes, success, and failure, true or false. It’s a really simple distribution, but worth knowing! In the example below we’re looking at the probability of rolling a 6 with a standard die.

If we roll a die many, many times, we should end up with a probability of rolling a 6, 1 out of every 6 times (or 16.7%) and thus a probability of not rolling a 6, in other words rolling a 1,2,3,4 or 5, 5 times out of 6 (or 83.3%) of the time! 

 

5. Discrete uniform distribution: All outcomes are equally likely 

Uniform distribution is represented by the function U(a, b), where a and b represent the starting and ending values, respectively. Like a discrete uniform distribution, there is a continuous uniform distribution for continuous variables.  

In statistics, uniform distribution refers to a statistical distribution in which all outcomes are equally likely. Consider rolling a six-sided die. You have an equal probability of obtaining all six numbers on your next roll, i.e., obtaining precisely one of 1, 2, 3, 4, 5, or 6, equaling a probability of 1/6, hence an example of a discrete uniform distribution. 

As a result, the uniform distribution graph contains bars of equal height representing each outcome. In our example, the height is a probability of 1/6 (0.166667). 

The drawbacks of this distribution are that it often provides us with no relevant information. Using our example of a rolling die, we get the expected value of 3.5, which gives us no accurate intuition since there is no such thing as half a number on a dice. Since all values are equally likely, it gives us no real predictive power. 

It is a distribution in which all events are equally likely to occur. Below, we’re looking at the results from rolling a die many, many times. We’re looking at which number we got on each roll and tallying these up. If we roll the die enough times (and the die is fair) we should end up with a completely uniform probability where the chance of getting any outcome is exactly the same 

 

6. Poisson distribution 

A Poisson Distribution is a discrete distribution similar to the Binomial Distribution (in that we’re plotting the probability of whole numbered outcomes) Unlike the other distributions we have seen however, this one is not symmetrical – it is instead bounded between 0 and infinity.  

For example, a cricket chirps two times in 7 seconds on average. We can use the Poisson distribution to determine the likelihood of it chirping five times in 15 seconds. A Poisson process is represented with the notation Po(λ), where λ represents the expected number of events that can take place in a period.

The expected value and variance of a Poisson process is λ. X represents the discrete random variable. A Poisson Distribution can be modeled using the following formula. 

The Poisson distribution describes the number of events or outcomes that occur during some fixed interval. Most commonly this is a time interval like in our example below where we are plotting the distribution of sales per hour in a shop. 

 

Conclusion: 

Data is an essential component of the data exploration and model development process. We can adjust our Machine Learning models to best match the problem if we can identify the pattern in the data distribution, which reduces the time to get to an accurate outcome.  

Indeed, specific Machine Learning models are built to perform best when certain distribution assumptions are met. Knowing which distributions, we’re dealing with may thus assist us in determining which models to apply. 

Data Science Dojo
Aadam Nadeem
| September 12

The Monte Carlo method is a technique for solving complex problems using probability and random numbers. Through repeated random sampling, Monte Carlo calculates the probabilities of multiple possible outcomes occurring in an uncertain process.  

Whenever you try to solve problems in the future, you make certain assumptions. For example, forecasting problems make certain assumptions like the cost of a particular item, the value of stocks, or electricity units used in the future. Since these problems try to predict an estimate of an unknown value based on historical data, there always exists inherent risk and uncertainty.  

The Monte Carlo simulation allows us to see all the possible outcomes of our decisions and assess risk, consequently allowing for better decision-making under uncertainty. 

This blog will walk through the famous Monty Hall problem, and how it can be solved using the Monte Carlo method using Python.  

Monty Hall problem 

In the Monty Hall problem, the TV show host Monty presents three doors to the participant. Behind one of the doors is a valuable prize like a car, while behind the others is a less valuable prize like a goat.  

Consider yourself to be one of the participants in the show. You choose one out of the three doors. Before opening your chosen door, Monty opens another door behind which would be one of the goats. Now you are left with two doors, behind one could be the car, and behind the other would be the other goat. 

Monty then gives you the option to either switch your answer to the other unopened door or stick to the original one.  

Is it in your favor to switch your answer to the other door? Well, probability says it is!  

Let’s see how: 

Initially, there are three unopen doors in front of you. The probability of the car being behind any of these doors is 1/3.  

 

Monte Carlo - Probability

 

Let’s say you decide to pick door #1 as the probability is the same (1/3) for each of these doors. In other words, the probability that the car is behind door #1 is 1/3, and the probability that it will be behind either door #2 or door #3 is 2/3. 

 

 

Monte Carlo - Probability

 

Monty is aware of the prize behind each door. He chooses to open door #3 and reveal a goat. He then asks you if you would like to either switch to door #2 or stick with door #1.  

 

Monte Carlo Probability

 

To solve the problem, let’s switch to Python and apply the Monte Carlo simulation. 

Solving with Python 

Initialize the 3 prizes

Python lists

 

Create python lists to store the probabilities after each game. We will play as many games as iterations input.  

 

Probability using Python

 

Monte Carlo simulation 

Before starting the game, we randomize the prizes behind each door. One of the doors will have a car behind it, while the other two will have a goat each. When we play a large number of games, all possible permutations get covered of prize distributions, and door choices get covered.  

 

Monte Carlo Simulations

 

Below is the code that decides if your choice was correct or not, and if switching would’ve been the correct move.  

 

Python code for Monte Carlo

 

 

 After playing each game, the winning probabilities are updated and stored in the lists. When all games have been played, we return the final values of each of the lists, i.e., winning by switching your choice and winning by sticking to your choice.  

 

calculating probabilities with Python

 

Get results

Enter your desired number of iterations (the higher the number, the more numbers of games will be played to approximate the probabilities). In the final step, plot your results.  

 

Probability - Python code

 

After running the simulation 1000 times, the probability that we win by always switching is 67.7%, and the probability that we win by always sticking to our choice is 32.3%. In other words, you will win approximately 2/3 times if you switch your door, and only 1/3 times if you stick to the original door. 

 

Probability results

 

Therefore, according to the Monte Carlo simulation, we are confident that it works to our advantage to switch the door in this tricky game. 

 

Ayesha Saleem - Digital content creator - Author
Ayesha Saleem
| September 9

In this blog, we will introduce you to the highly rated data science statistics books on Amazon. As you read the blog, you will find 5 books for beginners and 5 books for advanced-level experts. We will discuss what’s covered in each book and how it helps you to scale up your data science career. 

Statistics books

Advanced statistics books for data science 

1. Naked Statistics: Stripping the Dread from the Data – By Charles Wheelan 

Naked statistics by Charles Wheelan

The book unfolds the underlying impact of statistics on our everyday life. It walks the readers through the power of data behind the news. 

Mr. Wheelan begins the book with the classic Monty Hall problem. It is a famous, seemingly paradoxical problem using Bayes’ theorem in conditional probability. Moving on, the book separates the important ideas from the arcane technical details that can get in the way. The second part of the book interprets the role of descriptive statistics in crafting a meaningful summary of the underlying phenomenon of data. 

Wheelan highlights the Gini Index to show how it represents the income distribution of the nation’s residents and is mostly used to measure inequality. The later part of the book clarifies key concepts such as correlation, inference, and regression analysis explaining how data is being manipulated in order to tackle thorny questions. Wheelan’s concluding chapter is all about the amazing contribution that statistics will continue to make to solving the world’s most pressing problems, rather than a more reflective assessment of its strengths and weaknesses.  

2. Bayesian Methods For Hackers – Probabilistic Programming and Bayesian Inference, By Cameron Davidson-Pilon 

Bayesian methods for hackers

We mostly learn Bayesian inference through intensely complex mathematical analyses that are also supported by artificial examples. This book comprehends Bayesian inference through probabilistic programming with the powerful PyMC language and the closely related Python tools NumPy, SciPy, and Matplotlib. 

Davidson-Pilon focused on improving learners’ understanding of the motivations, applications, and challenges in Bayesian statistics and probabilistic programming. Moreover, this book brings a much-needed introduction to Bayesian methods targeted at practitioners. Therefore, you can reap the most benefit from this book if you have a prior sound understanding of statistics. Knowing about prior and posterior probabilities will give an added advantage to the reader in building and training the first Bayesian model.    

Read this blog if you want to learn in detail about statistical distributions

The second part of the book introduces the probabilistic programming library for Python through a series of detailed examples and intuitive explanations, with recent core developments and the popularity of the scientific stack in Python, PyMC is likely to become a core component soon enough. PyMC does have dependencies to run, namely NumPy and (optionally) SciPy. To not limit the user, the examples in this book will rely only on PyMC, NumPy, SciPy, and Matplotlib. This book is filled with examples, figures, and Python code that make it easy to get started solving actual problems.  

3. Practical Statistics for Data Scientists – By Peter Bruce and Andrew Bruce  

Practical statistics for data scientists

This book is most beneficial for readers that have some basic understanding of R programming language and statistics.  

The authors penned the important concepts to teach practical statistics in data science and covered data structures, datasets, random sampling, regression, descriptive statistics, probability, statistical experiments, and machine learning. The code is available in both Python and R. If an example code is offered with this book, you may use it in your programs and documentation.  

The book defines the first step in any data science project that is exploring the data or data exploration. Exploratory data analysis is a comparatively new area of statistics. Classical statistics focused almost exclusively on inference, a sometimes-complex set of procedures for drawing conclusions about large populations based on small samples.  

To apply the statistical concepts covered in this book, unstructured raw data must be processed and manipulated into a structured form—as it might emerge from a relational database—or be collected for a study.  

4. Advanced Engineering Mathematics by Erwin Kreyszig